For zero downtime deployment I use on my server dokku.
It has many features and is based on docker. You just install it on a server and then you add a git remote connection and then you just push and dokku makes all the magic ;)
When you have for example a Express server with a route like /myurl/test then you can create a CHECK with contains this route and during the startup dokku checks if this route is available.
It has more features just check it out :)